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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Forming Polynomials", "Polynomial Zeros", "Polynomial Degree" ], "new_concepts": [], "current_concepts": [ "Forming Polynomials", "Polynomial Zeros", "Polynomial Degree" ] } ## Set up the factored form of the polynomial $$ f(x) = 1 \cdot (x - (-1))(x - 1)(x - 8) = (x + 1)(x - 1)(x - 8) $$ ## Expand the linear factors $$ \begin{aligned} (x + 1)(x - 1) &= x^2 - 1\ f(x) &= (x^2 - 1)(x - 8) \end{aligned} $$ ## Distribute to find the final polynomial expression $$ f(x) = x^3 - 8x^2 - x + 8 $$ Form a polynomial whose zeros and degree are given. Zeros: $-1, 1, 8$; degree: 3 A polynomial with integer coefficients and a leading coefficient of 1 is $f(x) =$ $x^3 - 8x^2 - x + 8$. (Simplify your answer.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Forming Polynomials" ] }

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Explanation:

Set up the factored form of the polynomial

Expand the linear factors

Distribute to find the final polynomial expression

Answer: